For the most part, we would tell our Laptop users that it really didn't matter that much. If you are not going to be using it for a day, then turn it off. If you have it plugged in and will be using from time to time, leave it on. Hibernate should be an automatic thing it goes into, if you have 2000 or XP you can put it in standby mode, too. But be sure to save everythign first - some of the Dells in particular have problems in standby.
As for the that problem, have that checked! There are some models that just have a problem with heat, but in many cases either something is bad, or you are blocking the ventilation ports (happens a lot). All that heat is more dangerous to the internal components.
If you are having problems with short life on the batteries, you can try a couple of things. If you constantly drain the batteries a little and then have it plugged in for a long time, they often develop memories. They say the new batteries don't do this but that has not been our observation. The other problem we found was that some people never run it off the batteries, and the batteries don't seem to like that. It's a good idea to run it on the batteries for a while every once in a while. Batteries only have a lifespan of about 2-3 years anyway.